Finite Group Databasebeta

← #16733712

Subgroups — #16733712

Order 3328.

2130 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719530C211yesmaximal, characteristic, complemented
2#13125727C214yesmaximal, complemented
3#13125712C218yesmaximal, complemented
4#13125708C214yesmaximal, complemented
5#13124784C212yesmaximal, complemented
6#13124765C212yesmaximal, complemented
7#13124468C214yesmaximal, complemented
8#13124096C212yesmaximal, complemented
9#13124069C212yesmaximal, complemented
10#13123281C211yesmaximal, characteristic, complemented
11C13×#3449C211yesmaximal, characteristic
12C22×C13×#41C2211yescharacteristic
13C22×C13×#41C2211yescharacteristic
14C23×Dic26C2211yescharacteristic, complemented
15C13×#537C2214yes—
16C13×#536—21no—
17C13×#536C2212yes—
18C13×#535—21no—
19C13×#535C2212yes—
20C13×#534—21no—
21C13×#534—21no—
22C13×#532—22no—
23C13×#525—22no—
24C13×#525C2212yes—
25C13×#525C2212yes—
26C2×C13×#117C2211yescharacteristic
27#11719530—22no—
28#11719530—22no—
29#11719530C2214yes—
30#11719530C2214yescomplemented
31C2×#88205C2214yes—
32C2×#88198—21no—
33C2×#88190—21no—
34C2×#88189—21no—
35C2×#88189—21no—
36C2×#88180C2211yescharacteristic, complemented
37C2×#88178C2212yes—
38C2×#88178C2212yescomplemented
39C22×#6099C2212yescomplemented
40#11718900C2218yescomplemented
41#11718895C2214yes—
42#11718895C2214yescomplemented
43#11718893C2214yes—
44#11718889C2214yescomplemented
45#11718879C2214yescomplemented
46#11718868C2212yes—
47#11718868C2212yescomplemented
48#11718713C2218yescomplemented
49#11718712C2218yescomplemented
50#11718711C2218yescomplemented
51#11718706C2218yescomplemented
52#11718703C2218yescomplemented
53#11718702C2218yescomplemented
54#11718701C2218yescomplemented
55#11718696C2218yescomplemented
56#11718665C2214yescomplemented
57#11718657C2218yescomplemented
58#11718656C2214yescomplemented
59#11718632C2214yescomplemented
60#11718631C2214yescomplemented
61C2×C13×#41—21no—
62C2×C13×#41—21no—
63C2×C13×#41—44no—
64C2×C13×#41—44no—
65C2×C13×#41D412yes—
66C2×C13×#41D412yes—
67C2×C13×#41C2312yes—
68C2×C13×#41C2312yes—
69C2×C13×#41C2312yes—
70C2×C13×#41C2312yes—
71C22×Q8×C13D411yescharacteristic
72C22×Q8×C13C2311yescharacteristic
73C22×D4×C13—21no—
74C22×D4×C13D411yescharacteristic
75C22×D4×C13D411yescharacteristic
76C22×D4×C13D411yescharacteristic
77C22×D4×C13C2311yescharacteristic
78C23×C4×C13—21no—
79C23×C4×C13C2311yescharacteristic
80C23×C4×C13C2311yescharacteristic
81C23×C4×C13C2311yescharacteristic
82C23×Dic13—21no—
83C22×Dic26—21no—
84C22×Dic26—22no—
85C22×Dic26—44no—
86C22×Dic26C2312yes—
87C22×Dic26C2314yes—
88C13×C4⋊Q8—22no—
89C13×#127—22no—
90C13×#124—22no—
91C13×#124—22no—
92C13×#123—22no—
93C13×#123—22no—
94C13×#123—22no—
95C13×#123—22no—
96C13×C22⋊Q8—24no—
97C13×C22⋊Q8C2314yes—
98C13×C22⋊Q8C2314yes—
99C13×#121—24no—
100C13×#121—24no—
101C13×#121—24no—
102C13×#121C2314yes—
103C13×#121C2314yes—
104C13×#120—22no—
105C13×#120—22no—
106C13×#120—22no—
107C13×#120—22no—
108C4×D4×C13—24no—
109C4×D4×C13—24no—
110C13×#117—21no—
111C13×#117—21no—
112C13×#117C2312yes—
113C13×#117C2312yes—
114C2×C13×C4⋊C4C2312yes—
115C2×C13×C22⋊C4—21no—
116C2×C42×C13—21no—
117C2×C22⋊Dic13—22no—
118C2×C22⋊Dic13—22no—
119C2×C22⋊Dic13—21no—
120#88205—48no—
121#88198—24no—
122#88198—24no—
123C2×#6107C2318yes—
124#88190—24no—
125#88190—24no—
126#88189—24no—
127#88189—24no—
128#88189—24no—
129#88189—24no—
130C2×#6105C2318yes—
131#88180—22no—
132#88180C2314yes—
133C2×C4⋊Dic13C2312yes—
134#88178—22no—
135#88178—22no—
136#88178—24no—
137#88178C2314yes—
138#88178C2318yes—
139#88178C2314yes—
140C2×#6101—22no—
141C2×#6101—22no—
142C2×C4×Dic13—21no—
143C2×#6099—44no—
144C2×#6099C2314yes—
145C2×#6099C2314yes—
146C22×#1055C2314yes—
147#88075C2318yes—
148#88072C2318yes—
149C22⋊#1055C2314yes—
150C22⋊#1055C2318yes—
151C22⋊#1055C2314yes—
152#88050C2318yes—
153#88050C2318yes—
154#88048C2318yes—
155#88047C2318yes—
156#60754—131nomaximal, Sylow
157C24×C13—21no—
158C24×C13C2×D411yescharacteristic
159C24×C13C2×D411yescharacteristic
160C24×C13C2×D411yescharacteristic
161C13×#41—28no—
162C13×#41—44no—
163C13×#41—44no—
164C13×#41—42no—
165C13×#41—42no—
166C13×#41—42no—
167C13×#41—42no—
168C13×#41—44no—
169C13×#41—44no—
170C13×#41—44no—
171C2×Q8×C13—21no—
172C2×Q8×C13—42no—
173C2×Q8×C13—42no—
174C2×Q8×C13C2×D412yes—
175C2×Q8×C13C2×D412yes—
176C2×Q8×C13C2412yes—
177C2×D4×C13—22no—
178C2×D4×C13—22no—
179C2×D4×C13—24no—
180C2×D4×C13—24no—
181C2×D4×C13—21no—
182C2×D4×C13—42no—
183C2×D4×C13—42no—
184C2×D4×C13—42no—
185C2×D4×C13—42no—
186C2×D4×C13—44no—
187C2×D4×C13C2×D412yes—
188C2×D4×C13C2×D412yes—
189C2×D4×C13C2×D412yes—
190C2×D4×C13C2×D412yes—
191C2×D4×C13C2×D412yes—
192C2×D4×C13C2×D412yes—
193C2×D4×C13C2412yes—
194C22×C4×C13—24no—
195C22×C4×C13—24no—
196C22×C4×C13—24no—
197C22×C4×C13—21no—
198C22×C4×C13—22no—
199C22×C4×C13—22no—
200C22×C4×C13—22no—
201C22×C4×C13—44no—
202C22×C4×C13C2×D411yescharacteristic
203C22×C4×C13C2×D411yescharacteristic
204C22×C4×C13C2×D411yescharacteristic
205C22×C4×C13C2×D411yescharacteristic
206C22×C4×C13C2×D411yescharacteristic
207C22×C4×C13C2×D412yes—
208C22×C4×C13C2×D412yes—
209C22×C4×C13C2411yescharacteristic
210C22×C4×C13C2411yescharacteristic
211C22×C4×C13C2412yes—
212C22×C4×C13C2412yes—
213C22×C4×C13C2411yescharacteristic
214C22×Dic13—22no—
215C22×Dic13—22no—
216C22×Dic13—24no—
217C22×Dic13—44no—
218C2×Dic26—28no—
219C2×Dic26—22no—
220C2×Dic26—24no—
221C2×Dic26—44no—
222C2×Dic26—44no—
223C2×Dic26—44no—
224C2×Dic26—48no—
225C2×Dic26C2414yes—
226C13×C4⋊C4—22no—
227C13×C4⋊C4—24no—
228C13×C4⋊C4—24no—
229C13×C4⋊C4C2414yes—
230C13×C22⋊C4—22no—
231C13×C22⋊C4—22no—
232C13×C22⋊C4—24no—
233C13×C22⋊C4—24no—
234C13×C22⋊C4—24no—
235C13×C22⋊C4—24no—
236C42×C13—22no—
237C42×C13—22no—
238C22⋊Dic13—24no—
239C22⋊Dic13—28no—
240C22⋊Dic13—24no—
241C22⋊Dic13—24no—
242#6107—48no—
243#6105—48no—
244C4⋊Dic13—22no—
245C4⋊Dic13C2414yes—
246#6101—24no—
247#6101—28no—
248#6101—24no—
249C4×Dic13—24no—
250#6099—28no—
251#6099—44no—
252C2×#1055—44no—
253C2×#1055C2418yes—
254#3449D1311yescharacteristic
255C2×#536—131no—
256#3202—134no—
257#3201—138no—
258#3200—134no—
259#3095—132no—
260#3093—132no—
261#3056—134no—
262#3007—132no—
263#3004—132no—
264#2879—131no—
265C23×C13—22no—
266C23×C13—22no—
267C23×C13—22no—
268C23×C13—22no—
269C23×C13—42no—
270C23×C13—42no—
271C23×C13—42no—
272C23×C13—42no—
273C23×C13—82no—
274C23×C13C22×D411yescharacteristic
275C23×C13C22×D412yes—
276C23×C13C22×D411yescharacteristic
277C23×C13C22×D412yes—
278C23×C13C22×D411yescharacteristic
279Q8×C13—24no—
280Q8×C13—42no—
281Q8×C13—42no—
282Q8×C13—42no—
283D4×C13—24no—
284D4×C13—42no—
285D4×C13—42no—
286D4×C13—42no—
287D4×C13—42no—
288D4×C13—42no—
289D4×C13—42no—
290D4×C13—42no—
291D4×C13—44no—
292D4×C13—44no—
293D4×C13—84no—
294D4×C13—84no—
295C2×C4×C13—22no—
296C2×C4×C13—22no—
297C2×C4×C13—22no—
298C2×C4×C13—24no—
299C2×C4×C13—22no—
300C2×C4×C13—24no—
301C2×C4×C13—22no—
302C2×C4×C13—22no—
303C2×C4×C13—22no—
304C2×C4×C13—22no—
305C2×C4×C13—22no—
306C2×C4×C13—22no—
307C2×C4×C13—22no—
308C2×C4×C13—24no—
309C2×C4×C13—22no—
310C2×C4×C13—44no—
311C2×C4×C13—42no—
312C2×C4×C13—44no—
313C2×C4×C13—82no—
314C2×C4×C13C22×D412yes—
315C2×C4×C13C22×D411yescharacteristic
316C2×C4×C13C22×D412yes—
317C2×C4×C13C22×D411yescharacteristic
318C2×C4×C13C22×D411yescharacteristic
319C2×C4×C13H(5,𝔽2)14yes—
320C2×C4×C13C2511yescharacteristic
321C2×Dic13—24no—
322C2×Dic13—24no—
323C2×Dic13—48no—
324C2×Dic13—44no—
325C2×Dic13—44no—
326Dic26—28no—
327Dic26—48no—
328Dic26—44no—
329#1055—28no—
330C22×#41C2×D1311yescharacteristic
331C22×#41C2×D1311yescharacteristic
332C23×Q8—131no—
333C2×#137—134no—
334C22×M16—132no—
335#537C2×D1314yes—
336#536—21no—
337#536—134no—
338#536—134no—
339#536—262no—
340#536—262no—
341#536C2×D1312yes—
342#535—21no—
343#535C2×D1312yes—
344#534—21no—
345#534—21no—
346#532—22no—
347C2×C4⋊Q8—261no—
348C2×#124—261no—
349#525—22no—
350#525C2×D1312yes—
351#525C2×D1312yes—
352C2×#123—261no—
353C2×#123—261no—
354C2×C22⋊Q8—132no—
355C2×C22⋊Q8—132no—
356C2×#117—131no—
357C2×#117C2×D1311yescharacteristic
358#485—138no—
359#485—138no—
360#484—134no—
361#483—134no—
362#482—138no—
363#470—138no—
364#468—138no—
365#467—138no—
366#465—138no—
367#452—138no—
368#451—134no—
369#451—138no—
370#450—134no—
371#448—138no—
372#428—134no—
373#419—134no—
374#417—134no—
375#408—134no—
376#408—134no—
377#407—132no—
378#407—132no—
379C22×C13—22no—
380C22×C13—22no—
381C22×C13—24no—
382C22×C13—24no—
383C22×C13—22no—
384C22×C13—22no—
385C22×C13—21no—
386C22×C13—21no—
387C22×C13—22no—
388C22×C13—44no—
389C22×C13—44no—
390C22×C13—42no—
391C22×C13—82no—
392C22×C13—82no—
393C22×C13—84no—
394C22×C13#53412yes—
395C22×C13C23×D411yescharacteristic
396C4×C13—22no—
397C4×C13—21no—
398C4×C13—21no—
399C4×C13—22no—
400C4×C13—44no—
401C4×C13—42no—
402C4×C13#53412yes—
403C4×C13C2×H(5,𝔽2)12yes—
404Dic13—44no—
405Dic13—44no—
406C2×#41—21no—
407C2×#41—21no—
408C2×#41—44no—
409C2×#41—44no—
410C2×#41C22⋊D1312yes—
411C2×#41C22⋊D1312yes—
412C2×#41C22×D1312yes—
413C2×#41C22×D1312yes—
414C2×#41C22×D1312yes—
415C2×#41C22×D1312yes—
416C22×Q8—132no—
417C22×Q8—134no—
418C22×Q8—261no—
419C22×Q8—262no—
420C22×Q8—524no—
421C22×Q8C22⋊D1311yescharacteristic
422C22×Q8C22×D1311yescharacteristic
423C22×D4—21no—
424C22×D4C22⋊D1311yescharacteristic
425C22×D4C22⋊D1311yescharacteristic
426C22×D4C22⋊D1311yescharacteristic
427C22×D4C22×D1311yescharacteristic
428C23×C4—21no—
429C23×C4—261no—
430C23×C4C22×D1311yescharacteristic
431C23×C4C22×D1311yescharacteristic
432C23×C4C22×D1311yescharacteristic
433#137—528no—
434C2×Q16—138no—
435C2×SD16—138no—
436C2×M16—134no—
437C2×M16—134no—
438C2×M16—524no—
439C22×C8—134no—
440C4⋊Q8—22no—
441C4⋊Q8—264no—
442C4⋊Q8—264no—
443#127—22no—
444#124—22no—
445#124—22no—
446#124—264no—
447#124—264no—
448#123—22no—
449#123—22no—
450#123—22no—
451#123—22no—
452#123—264no—
453#123—264no—
454#123—264no—
455#123—264no—
456C22⋊Q8—24no—
457C22⋊Q8—134no—
458C22⋊Q8—134no—
459C22⋊Q8—138no—
460C22⋊Q8—262no—
461C22⋊Q8—262no—
462C22⋊Q8—264no—
463C22⋊Q8C22×D1314yes—
464C22⋊Q8C22×D1314yes—
465#121—24no—
466#121—24no—
467#121—24no—
468#121C22×D1314yes—
469#121C22×D1314yes—
470#120—22no—
471#120—22no—
472#120—22no—
473#120—22no—
474C4×D4—24no—
475C4×D4—24no—
476#117—21no—
477#117—21no—
478#117—134no—
479#117—262no—
480#117C22×D1312yes—
481#117C22×D1312yes—
482C2×C4⋊C4—132no—
483C2×C4⋊C4—262no—
484C2×C4⋊C4—262no—
485C2×C4⋊C4C22×D1312yes—
486C2×C22⋊C4—21no—
487C2×C22⋊C4—261no—
488C2×C22⋊C4—262no—
489C2×C22⋊C4—262no—
490C2×C42—21no—
491C2×C42—261no—
492#107—138no—
493#106—138no—
494#103—138no—
495#103—138no—
496#103—138no—
497#102—138no—
498C22⋊C8—138no—
499C22⋊C8—134no—
500C22⋊C8—134no—
501C2×C13—22no—
502C2×C13—42no—
503C2×C13—41no—
504C2×C13—42no—
505C2×C13—41no—
506C2×C13—82no—
507C2×C13C2×#53411yescharacteristic
508C2×C13#358612yes—
509C24—21no—
510C24C2×C22⋊D1311yescharacteristic
511C24C2×C22⋊D1311yescharacteristic
512C24C2×C22⋊D1311yescharacteristic
513#41—28no—
514#41—44no—
515#41—44no—
516#41—42no—
517#41—42no—
518#41—42no—
519#41—42no—
520#41—44no—
521#41—44no—
522#41—44no—
523C2×Q8—21no—
524C2×Q8—42no—
525C2×Q8—42no—
526C2×Q8—134no—
527C2×Q8—262no—
528C2×Q8—264no—
529C2×Q8—268no—
530C2×Q8—524no—
531C2×Q8—524no—
532C2×Q8—524no—
533C2×Q8—528no—
534C2×Q8C2×C22⋊D1312yes—
535C2×Q8C2×C22⋊D1312yes—
536C2×Q8C23×D1312yes—
537C2×D4—22no—
538C2×D4—22no—
539C2×D4—24no—
540C2×D4—24no—
541C2×D4—21no—
542C2×D4—42no—
543C2×D4—42no—
544C2×D4—42no—
545C2×D4—42no—
546C2×D4—44no—
547C2×D4C2×C22⋊D1312yes—
548C2×D4C2×C22⋊D1312yes—
549C2×D4C2×C22⋊D1312yes—
550C2×D4C2×C22⋊D1312yes—
551C2×D4C2×C22⋊D1312yes—
552C2×D4C2×C22⋊D1312yes—
553C2×D4C23×D1312yes—
554C22×C4—24no—
555C22×C4—24no—
556C22×C4—24no—
557C22×C4—21no—
558C22×C4—22no—
559C22×C4—22no—
560C22×C4—22no—
561C22×C4—44no—
562C22×C4—262no—
563C22×C4—262no—
564C22×C4—264no—
565C22×C4—524no—
566C22×C4C2×C22⋊D1311yescharacteristic
567C22×C4C2×C22⋊D1311yescharacteristic
568C22×C4C2×C22⋊D1311yescharacteristic
569C22×C4C2×C22⋊D1311yescharacteristic
570C22×C4C2×C22⋊D1311yescharacteristic
571C22×C4C2×C22⋊D1312yes—
572C22×C4C2×C22⋊D1312yes—
573C22×C4C23×D1311yescharacteristic
574C22×C4C23×D1312yes—
575C22×C4C23×D1311yescharacteristic
576C22×C4C23×D1312yes—
577C22×C4C23×D1311yescharacteristic
578Q16—528no—
579SD16—528no—
580M16—268no—
581M16—524no—
582C2×C8—138no—
583C2×C8—524no—
584C4⋊C4—22no—
585C4⋊C4—24no—
586C4⋊C4—24no—
587C4⋊C4—134no—
588C4⋊C4—262no—
589C4⋊C4—264no—
590C4⋊C4—268no—
591C4⋊C4—264no—
592C4⋊C4C23×D1314yes—
593C22⋊C4—22no—
594C22⋊C4—22no—
595C22⋊C4—24no—
596C22⋊C4—24no—
597C22⋊C4—24no—
598C22⋊C4—24no—
599C22⋊C4—264no—
600C22⋊C4—264no—
601C22⋊C4—268no—
602C22⋊C4—264no—
603C42—22no—
604C42—22no—
605C42—264no—
606C13#6075411yescharacteristic, complemented, Sylow
607C23—22no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—82no—
616C23C22×C22⋊D1311yescharacteristic
617C23C22×C22⋊D1312yes—
618C23C22×C22⋊D1312yes—
619C23C22×C22⋊D1311yescharacteristic
620C23C22×C22⋊D1311yescharacteristic
621Q8—24no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—268no—
626Q8—528no—
627Q8—524no—
628D4—24no—
629D4—42no—
630D4—42no—
631D4—42no—
632D4—42no—
633D4—42no—
634D4—42no—
635D4—42no—
636D4—44no—
637D4—44no—
638D4—84no—
639D4—84no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—82no—
659C2×C4—264no—
660C2×C4—264no—
661C2×C4—524no—
662C2×C4—528no—
663C2×C4—524no—
664C2×C4#8825114yes—
665C2×C4C22×C22⋊D1311yescharacteristic
666C2×C4C22×C22⋊D1312yes—
667C2×C4C22×C22⋊D1312yes—
668C2×C4C22×C22⋊D1311yescharacteristic
669C2×C4C22×C22⋊D1311yescharacteristic
670C2×C4C24×D1311yescharacteristic
671C8—268no—
672C22—22no—
673C22—22no—
674C22—24no—
675C22—24no—
676C22—22no—
677C22—21no—
678C22—22no—
679C22—21no—
680C22—22no—
681C22—44no—
682C22—44no—
683C22—42no—
684C22—82no—
685C22—82no—
686C22—84no—
687C22#1171950912yes—
688C22C23×C22⋊D1311yescharacteristic
689C4—21no—
690C4—21no—
691C4—22no—
692C4—22no—
693C4—44no—
694C4—42no—
695C4—524no—
696C4—524no—
697C4#1171950912yes—
698C4C2×#8825112yes—
699C2—22no—
700C2—42no—
701C2—41no—
702C2—41no—
703C2—42no—
704C2—82no—
705C2C2×#1171950911yescharacteristic
706C2#1312962212yes—
subgroupG/H
centerC22C23×C22⋊D13
commutator, cosocleC2×C4×C13C25
FrattiniC2×C4C24×D13
Fitting, prime radicalC13×#3449C2
socleC22×C13C23×D4
hypercenter#3449D13
hypocenterC13#60754
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×#41 ◁ #3449

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#11719530C2semidirect
2. #13125727C2semidirect
3. #13125712C2semidirect
4. #13125708C2semidirect
5. #13124784C2semidirect
6. #13124765C2semidirect
7. #13124468C2semidirect
8. #13124096C2semidirect
9. #13124069C2semidirect
10. #13123281C2semidirect
14. C23×Dic26C22semidirect
30. #11719530C22semidirect
36. C2×#88180C22semidirect
38. C2×#88178C22semidirect
39. C22×#6099C22semidirect
40. #11718900C22semidirect
42. #11718895C22semidirect
44. #11718889C22semidirect
45. #11718879C22semidirect
47. #11718868C22semidirect
48. #11718713C22semidirect
49. #11718712C22semidirect
50. #11718711C22semidirect
51. #11718706C22semidirect
52. #11718703C22semidirect
53. #11718702C22semidirect
54. #11718701C22semidirect
55. #11718696C22semidirect
56. #11718665C22semidirect
57. #11718657C22semidirect
58. #11718656C22semidirect
59. #11718632C22semidirect
60. #11718631C22semidirect
606. C13#60754semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117195302
#131257272
#131257122
#131257082
#131247842
#131247652
#131244682
#131240962
#131240692
#131232812
C13×#34492
#6075413